Acoustics 2

Periodic Waves

terms: periodic waveforms – cycle – period – wavelength – frequency – Hertz

Most of the sounds used in music have periodic waveforms – repeating patterns of air pressure change – that produce a discernible pitch. Each repeating segment is called a cycle. The duration of time each cycle takes to complete the pattern is called a period (measured in seconds). The acoustic energy produced by the air pressure change in each cycle takes a certain distance to propagate through the air called the wavelength (measured in feet per cycle).

We mostly refer to periodic waveforms in terms of frequency – how fast cycles repeat. Frequency is measured in cycles per second or Hertz (Hz), named after Heinrich Hertz, the German physicist who researched electromagnetic waves.

While we tend to think of pitched sounds as “high/low” – in reference to the number of cycles per second – we could easily think in terms of “fast/slow” (period) or “short/long” (wavelength).

PERIOD = time it takes to complete one cycle
WAVELENGTH = distance required to complete one cycle
FREQUENCY = number of cycles per second

If a cycle is repeating 10 times per second, or has a frequency of 10 Hz, the period is 1/10 of a second, and the wavelength is ≈113 feet.

Play different pitches on the keyboard below. Observe the changing values for frequency, period, and wavelength. The sound you hear is a sine wave.

As the frequency rises, what happens to the period and wavelength?

Phase

Phase describes a particular point in the cycle of a waveform (measured in degrees). It is not usually an audible characteristic of a single wave, but becomes an important factor in the interaction of one wave with another (and/or their partials). We’ll stay out of the weeds and leave the details for some other time. But here’s a link for the mathematically curious. Here are a few examples of these interactions in the video below. The red lines indicate 2 cycles at 261 Hz. “E” is the resulting signal from the combined sine tones.

Constructive Interference A+B: 2 signals with the same frequency and phase when combined increase the amplitude.

Destructive Interference A+C: 2 signals with the same frequency but the 2nd is 180° out of phase when combined completely cancel each other out. Also known as phase cancellation.

Beating or Beat Frequency A+D: 2 signals with very close frequencies produce a pulsation at a rate equal to the difference between the two frequencies. Here the difference is “1” so the pulsation occurs every second.

Two examples of phase cancellation or destructive interference are below. The first example is white noise. The second track is an inverted copy of the white noise track above. Notice how the overall volume goes down when the second track fades in. The second example is the same process, but with an acoustic guitar. Here the guitar completely disappears, but notice the track meters, they are still showing sound.

Here’s a good analogy for phase cancellation: Imagine two people pushing against a box in opposite directions with the exact same force. Energy is still being used, but the result is zero movement. The same thing happens with an audio signal. When one wave compresses while another at the same frequency and amplitude

Waveform Displays

terms: waveform display – oscilloscope – range of human hearing

It is often useful to visualize waveforms. In music software, we often encounter waveform displays that plot the varying amplitude of the waveform over time. The oscilloscope below is one such display. It shows a short section of the wave changing in real time within a window with a fixed duration. The range of human hearing is 20 Hz to 20,000 Hz (often expressed as 20 kHz, or kilohertz). As we age, some high-frequency hearing is lost.

Thinking about the relationship between “frequency” and “period” how would you expect the waveform display to change the frequency is adjusted?

If the slider is all the way on the left, how many periods fit within the waveform display? At what frequency will there be exactly twice as many periods?

Fourier Analysis

terms: partial – spectrum – FFT – spectrogram – sonogram

So far, we’ve been listening to and looking at sine waves which have a single frequency. Now let’s look at more complex waveforms from a piano, guitar, and sitar.

Jean Baptiste Fourier, an 18th-century French mathematician, showed that any complex periodic waveform can be understood as the sum of a series of related sine waves of various frequencies, phases, and amplitudes. Each of these sine waves is called a partial. The complete set is known as the spectrum of the sound. The mathematical function that performs this analysis is the Fast Fourier Transform, or FFT.

We use a type of graph, a 2D spectrogram, to see these multiple sine waves. The graph shows the amplitudes (y-axis) of all simultaneously sounding sine-wave partials and indicates their frequencies (x-axis). When the note plays, the vertical blue spikes in the graph represent the individual partials. The leftmost spike is the fundamental, or the note we perceive as the pitch.

The sonogram on the right is another way of graphing frequency over time. Here amplitudes of each frequency are indicated with shading – darker lines mean higher amplitudes.

In these graphs, we are only looking at frequencies below 5000 Hz where most of the frequency content is in these sounds, even though the audible sound spectrum continues up to 20 kHz.

Harmonic Partials

terms: fundamental frequency – timbre – harmonic

Notice that the guitar partials are evenly spaced across the spectrum. Each adjacent partial is separated by about 260 Hz. Sounds with a clear pitch have uniform spacing of partials, these partials are mostly periodic frequencies.

Watch the partials in the graph. The left most partial is the fundamental frequency. This is the frequency we normally think of as the pitch of a note. In this case, about 260 Hz or middle C. The partials (sometimes called harmonics) perceptually fuse together to create the tone color or timbre of the sound. We don’t generally hear sounds as a collection of partials, but as a single sound. There are some notable exceptions (think about the sound of a large bell).

We number partials starting with the fundamental as 1. The frequency of a harmonic partial is equal to the fundamental frequency x the partial number (260 * 1, 260 * 2, 260 * 3, 260 * 4, etc.). This relationship between partials is an expression of the harmonic series. Here is a great website that demonstrates the harmonic series as divisions of a string.

Additive Synthesis

If the Fast Fourier Transform allows us to take any complex waveform and reduce it to its component sine waves, then we should be able to combine sine waves to produce a complex waveform. This process – the Inverse Fourier Transform – is called Additive Synthesis.

Start with a fundamental frequency of 261 Hz. To it we will add the third, fifth, and seventh partials at 783, 1305, and 1827 Hz. The waveform plots (A, B, C, D) combine for form the composite waveform (E) that we hear. Adjust the volume knobs next to each waveform plot to mix the amplitudes of the partials. The red lines indicate cycles at 261 Hz.

Wave B vibrates 3 times faster than wave A, producing 3 cycles for every 1 of A.
Wave C vibrates 5 times faster than wave A, producing 5 cycles for every 1 of A.
Wave D vibrates 7 times faster than wave A, producing 3 cycles for every 1 of A.
By turning down B and turning up C and D respectively, watch how the wave rides up and down the slower moving wave A.

Inharmonic Partials

Many sounds that we use in music have a spectrum made up of only harmonically related partials: violin, trumpet, oboe, piano, etc. Their partials are integer (i.e. whole number) multiples of the fundamental frequency. Other sounds – especially percussion instruments like cymbals and bells – have partials that are not integer multiples of the fundamental. Instead these sounds feature inharmonic partials.

For example, the bell sound below has a fundamental of about 200 Hz. That frequency isn’t the loudest sounding partial, so our sense of a central pitch is distorted. Several of the prominent partial frequencies are labelled below. Some of these partials have a frequency close to a harmonic partial of 200 Hz, but slightly “out of tune.” There are also many more partials with lower amplitudes clustering around these higher peaks.

The combination of a fundamental with a low amplitude relative to higher partials, prominent partials near their harmonic equivalents, and dense clusters of lower amplitude partials that fade out quickly are properties of this quintessential “bell” sound.